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#############################################################################
##
#W matrix.gi GAP4 package `Utils' Pedro A. Garcia-Sanchez
##
#Y Copyright (C) 2023, The GAP Group
InstallMethod( DirectSumBlock, "for a matrix",
[ IsMatrixOrMatrixObj, IsInt, IsList, IsList ], 0,
function( M, level, blocks, answer )
local nrM, ncM, nM, nr0, nc0, found, nlzr, nlzc, ntzr, ntzc, nlz, ntz,
B, nrB, ncB, nB, nr, nc, i, j, k, i0, j0, nzr, nzc, rest, sub,
bls, nrbls, ncbls, pr1, diag0, up0, low0, imax, jmax,
len1, ok, ans;
nrM := NumberRows( M );
ncM := NumberColumns( M );
nM := Minimum( [ nrM, ncM ] );
## deal with the diagonal case first
if IsDiagonalMatrix( M ) then
bls := List( [1..nM], i -> [ [ M[i,i] ] ] );
if ( ncM > nM ) then
sub := ExtractSubMatrix( M, [nM], [nM..ncM] )[1];
bls[nM] := [ sub ];
fi;
if ( nrM > nM ) then
sub := ExtractSubMatrix( M, [nM..nrM], [nM] )[1];
bls[ nM] := [ sub ];
fi;
blocks[1] := Concatenation( blocks[1], bls );
ans := Concatenation( blocks[1], blocks[2] );
if not ( ans in answer ) then
Add( answer, ans );
fi;
return 0;
fi;
## check for leading or trailing rows or columns of zeros
nlzr := 0;
found := false;
i := 0;
while not found do
i := i+1;
if ForAll( [1..ncM], j -> M[i,j] = 0 ) then
nlzr := nlzr + 1;
else
found := true;
fi;
od;
nlzc := 0;
found := false;
j := 0;
while not found do
j := j+1;
if ForAll( [1..nrM], i -> M[i,j] = 0 ) then
nlzc := nlzc + 1;
else
found := true;
fi;
od;
ntzr := 0;
found := false;
i := nrM + 1;
while not found do
i := i-1;
if ForAll( [1..ncM], j -> M[i,j] = 0 ) then
ntzr := ntzr + 1;
else
found := true;
fi;
od;
ntzc := 0;
found := false;
j := ncM + 1;
while not found do
j := j-1;
if ForAll( [1..nrM], i -> M[i,j] = 0 ) then
ntzc := ntzc + 1;
else
found := true;
fi;
od;
## add any leading or trailing blocks [ [0] ]
nlz := Minimum( [ nlzr, nlzc ] );
if ( nlz > 0 ) then
blocks[1] := Concatenation( blocks[1],
ListWithIdenticalEntries( nlz, [ [0] ] ) );
fi;
ntz := Minimum( [ ntzr, ntzc ] );
if ( ntz > 0 ) then
blocks[2] := Concatenation(
ListWithIdenticalEntries( ntz, [ [0] ] ), blocks[2] );
fi;
B := ExtractSubMatrix( M, [nlz+1..nrM-ntz], [nlz+1..ncM-ntz] );
nrB := NumberRows( B );
ncB := NumberColumns( B );
nB := Minimum( [nrB, ncB] );
bls := [ ];
if ( nrB = 1 ) or ( ncB = 1 ) then
Add( blocks[1], B );
ans := Concatenation( blocks[1], blocks[2] );
if not ( ans in answer ) then
Add( answer, ans );
fi;
return 0;
fi;
## find first non-zero entry in column 1
found := false;
pr1 := 0;
while ( not found ) and ( pr1 < nrB-1 ) do
pr1 := pr1 + 1;
found := B[pr1,1] <> 0;
od;
## determine ranges for i0 and j0
diag0 := nB+1;
i := nB+1;
while ( i > 1 ) do
i := i-1;
if ( B[i,i] <> 0 ) then
diag0 := i;
fi;
od;
up0 := [diag0,diag0];
j := 1;
found := false;
while ( not found ) and ( j < ncB ) do
j := j+1;
k := Minimum( [ j-1, nrB ] );
for i in [1..k] do
if ( B[i,j] <> 0 ) then
up0 := [i,j];
found := true;
fi;
od;
od;
low0 := [diag0,diag0];
i := 1;
found := false;
while ( not found ) and ( i < nrB ) do
i := i+1;
k := Minimum( [ i-1, ncB ] );
for j in [1..k] do
if ( B[i,j] <> 0 ) then
low0 := [i,j];
found := true;
fi;
od;
od;
imax := diag0;
jmax := diag0;
if ( up0[2] < low0[2] ) then
jmax := low0[2] - 1;
fi;
if ( low0[1] < up0[1] ) then
imax := up0[1] - 1;
fi;
if ( up0[1] < low0[1] ) then
imax := Minimum( imax, low0[1] - 1 );
fi;
if ( low0[2] < up0[2] ) then
jmax := Minimum( jmax, up0[2] - 1 );
fi;
if ( low0[1] > nrB ) then
imax := up0[1];
fi;
if ( up0[1] > ncM ) then
jmax := low0[2];
fi;
i0 := 0;
while ( i0 < imax ) do
i0 := i0 + 1;
i := i0;
j0 := 0;
while ( j0 < jmax ) do
found := false;
j0 := j0 + 1;
if ( j0 <= pr1 ) then
j := j0;
rest := MutableCopyMatrix( B );
nr := NumberRows( rest );
nc := NumberColumns( rest );
i := i0;
j := j0;
nzr := true;
nzc := true;
while nzr or nzc do
nzr := ForAny( [1..i],
i1 -> ForAny( [j+1..nc], j1->rest[i1][j1]<>0 ) );
nzc := ForAny( [i+1..nr],
i1 -> ForAny( [1..j], j1->rest[i1][j1]<>0 ) );
if nzr then
j := j+1;
fi;
if nzc then
i := i+1;
fi;
od;
bls := List( [1..i], i1 -> rest[i1]{[1..j]} );
## check if we have filled all columns and rows
nzc := ntzc - ntz;
nrbls := NumberRows( bls );
ncbls := NumberColumns( bls );
if ( nzc > 0 ) and ( ncbls + nzc = ncB ) then
## add zero columns at the end of the block
j := j + nzc;
bls := List( [1..i], i1 -> rest[i1]{[1..j]} );
fi;
nzr := ntzr - ntz;
if ( nzr > 0 ) and ( nrbls + nzr = nrB ) then
## add zero rows at the end of the block
i := i + nzr;
bls := List( [1..i], i1 -> rest[i1]{[1..j]} );
fi;
Add( blocks[1], bls );
len1 := Length( blocks[1] );
rest := List( [i+1..nr], i1 -> rest[i1]{[j+1..nc]} );
if ( rest <> [ ] ) then
ok := DirectSumBlock( rest,level+1,blocks,answer );
else
ans := Concatenation( blocks[1], blocks[2] );
if not ( ans in answer ) then
Add( answer, ans );
fi;
fi;
blocks := [ blocks[1]{[1..len1-1]}, blocks[2] ];
fi;
od;
od;
return Length( answer );
end );
InstallMethod( DirectSumDecompositionMatrices, "for a matrix",
[ IsMatrixOrMatrixObj ], 0,
function( M )
local L, blocks, answer, a, ma;
blocks := [ [], [] ];
answer := [ ];
L := DirectSumBlock( M, 1, blocks, answer );
## check that the answers are correct
for a in answer do
ma := DirectSumMat( a );
if not ( ma = M ) then
Display( ma );
Error( "direct sum of decomposition <> original matrix" );
fi;
od;
return answer;
end );
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